While there were ancient speculations and mentions of tiny particles being the ultimate constituent of matter- there is no surviving explanation of the properties of matter in terms of the properties of such atoms ( or being widely known).
The modern conception of atoms and molecules starts with Dalton's proposals made around 1820 about atoms and molecules combining to form many compounds ( mostly inorganic compounds).
This was taken to another level in explaining the observed properties of gases starting with Avogadro's hypothesis about all gases with a mass equal to their molecular weight having the same number of molecules, followed by the kinetic theory of gases in its non ideal form( that is considering the size and inter molecular forces ) in the mid 1800s.
The expansion of chemistry, and the researches into atomic and molecular spectra and electrical phenomena in gases under low pressure were all explained by improvements in the understanding of the how atoms are able to interact with each other to give rise to various observed phenomena was a continuous process till the 1950s.
As to what constitutes a definitive proof of the existence of atoms and molecules- is something that could vary from person to person.
For me, the ability to create materials or phenomena not present in nature by applying the principles of atomic structure is proof enough.
So , starting from relatively simple polymers like polyethylene/ polyamides( nylon) to complex hormones based on permitted ( under the principles of atomic structure and combination) chemical reactions ; the invention of Lasers / LED lights / ability to see images of the human body by MRI/ PET scans are excellent examples of the correctness of our current models of atomic structure.